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Theorem ax11-pm 32497
Description: Proof of ax-11 2031 similar to PM's proof of alcom 2034 (PM*11.2). For a proof closer to PM's proof, see ax11-pm2 32501. Axiom ax-11 2031 is used in the proof only through nfa2 2037. (Contributed by BJ, 15-Sep-2018.) (Proof modification is discouraged.)
Assertion
Ref Expression
ax11-pm (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)

Proof of Theorem ax11-pm
StepHypRef Expression
1 2sp 2054 . . 3 (∀𝑥𝑦𝜑𝜑)
21gen2 1720 . 2 𝑦𝑥(∀𝑥𝑦𝜑𝜑)
3 nfa2 2037 . . 3 𝑦𝑥𝑦𝜑
4 nfa1 2025 . . 3 𝑥𝑥𝑦𝜑
53, 42stdpc5 32494 . 2 (∀𝑦𝑥(∀𝑥𝑦𝜑𝜑) → (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑))
62, 5ax-mp 5 1 (∀𝑥𝑦𝜑 → ∀𝑦𝑥𝜑)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1478
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-10 2016  ax-11 2031  ax-12 2044
This theorem depends on definitions:  df-bi 197  df-or 385  df-ex 1702  df-nf 1707
This theorem is referenced by: (None)
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